Exact conservation as selection principle: discrete exterior calculus for the incompressible Navier-Stokes and Euler equations

Abstract

We formulate a new discrete-exterior calculus based discretisation of the incompressible Euler and Navier-Stokes equations that preserves the geometric structure of the continuum, and establish a rigorous convergence and structure theory for a new discretisation. The discretisation operates on prismatic Delaunay-Voronoi meshes over closed Riemannian manifolds. The geometry of Euler and Navier-Stokes equations is maintained via a discrete Lie derivative that is built from an extrusion-based contraction for the nonlinear term in vector-invariant form. Conservation of energy and Kelvin circulation links the discrete scheme to the continuum: at the discrete level, energy conservation is a stability property, and in the vanishing-resolution limit it becomes both a constructive route into the conservative weak-solution theory of the continuum equations and a selection principle on the limits the scheme can reach. This correspondence appears in four regimes. \emph{Smooth solutions}: convergence at rate $\mathcal{O}(h^{\min (r_{\mathrm{rec}},r_{\star })}|\log h|)$ in dimensions $d=2,3$, uniformly in viscosity $\hat{I}½\ge 0$; first order on general meshes, second order under centroid proximity and reconstruction symmetry. \emph{Leray-Hopf weak regime}: subsequential $L^2$ limits of the discrete Navier-Stokes system are weak solutions of the viscous equations. \emph{Inviscid measure-valued regime}: limits are conservative measure-valued Euler solutions, with concentration defect vanishing above the Onsager threshold $\hat{I}\pm >1/3$ provided the discrete solutions admit a uniform $C^{0,\hat{I}\pm }$ bound; the scheme reaches the energy-conserving side of the Onsager landscape but not the dissipative side. \emph{Dissipative regime}: no subsequence converges to an energy-dissipating Euler solution at any Hölder regularity, an exclusion that follows from discrete energy conservation.